The Mathematical Principles Underlying Newton's Principia Mathematica:

TLDR: Newman's Principia as discussed by the authors is a difficult book to understand and it has been criticised for being difficult to understand by a limited number of experts, such as Dr William Derham, who pointed out that the logical structure of Newton's book is slipshod, its level of verbal fluency none too high, its arguments unnecessarily diffuse and repetitive, and its very content on occasion markedly irrelevant to its professed theme.

Abstract: On 18 July 1733, half a dozen years after Isaac Newton's death, Dr William Derham (a close friend during his last years) observed that \"S' Is[aac] ... abhorred all Contests.... And for this reason, mainly to avoid being baited by little Smatterers in Mathematicks, he told me, he designedly made his Principia abstruse; but yet so as to be understood by able Mathematicians, who he imagined, by comprehending his Demonstrations, would concurr with him in his Theory\".\" Forty years before, as Newton passed unseeingly by in the street at Cambridge, a nameless undergraduate had remarked sotto voce: \"There goes the man that writt a book that neither he nor anybody else understands\".\" Evidently, if it had been Newton's intention in the 1680s to make his mathematical world-view impossibly difficult for all but a tightly restricted elite to comprehend, in this one case at least he succeeded only too well. But was it? When we go behind such hearsay and anecdote, we will find that there is no trustworthy documentary evidence that Newton did deliberately contrive to render his Mathematical principles of natural philosophy more esoteric and impenetrable than he need have done. No one would deny that this ikon of scientific history is far from easy to read. Quite bluntly, the logical structure of Newton's book is slipshod, its level of verbal fluency none too high, its arguments unnecessarily diffuse and repetitive, and its very content on occasion markedly irrelevant to its professed theme: the theory of bodies in motion. But these faults are far from intentional and can largely be excused by the very rapidity with which the Principia was written-in little more than two years from the autumn of 16844and its author's distinct lack of talent for writing in a popular way. Ofsuch weaknesses no one was more conscious than Newton himself: indeed, we now know that in the early 1690s, soon after his book appeared, he reluctantly contemplated a grand revision of his work which he never found time and energy to irnplement.! In default we must suffer the crudities of the text as we have it in order to master the Principia's complex mathematical and scientific content. To attain this understanding there is no royal road which can bypass its conceptual difficulties, no novice's path which will soften its mathematical rigours. Newton himself tried to lighten the heavy load of preparatory learning for such acquaintances as Trinity College's guiding genius, Richard Bentley, who sought to achieve a limited understanding of merely the basic propositions of the Principia, but he succeeded none too well in his altruistic purpose. Having given Bentley the eminently sensible advice that \"At') first perusal of my Book it's enough if you understand y. Propositions WI\" some of') Demonvtrations we\" are easier than the rest ... [then] pass on to v: 3 Book & when you 'lee the design of that you may turn back to such Propositions as you shall have a desire to know\", Newton was then ineluctably led to compile for him in 1691 a formidable mathematical reading list\" which of necessity referred much more to recent works in geometrical and infinitesimal analysis than to traditional algebra and classical geometry. The Principia was-